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Black Model
The Black model (sometimes known as the Black-76 model) is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing options on future contracts, bond options, interest rate cap and floors, and swaptions. It was first presented in a paper written by Fischer Black in 1976. Black's model can be generalized into a class of models known as log-normal forward models, also referred to as LIBOR market model. The Black formula The Black formula is similar to the Black–Scholes formula for valuing stock options except that the spot price of the underlying is replaced by a discounted futures price F. Suppose there is constant risk-free interest rate ''r'' and the futures price ''F(t)'' of a particular underlying is log-normal with constant volatility ''σ''. Then the Black formula states the price for a European call option of maturity ''T'' on a futures contract with strike price ''K'' and delivery date ''T (with T' \geq T) is : c = e^ N(d_1) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Futures Contract
In finance, a futures contract (sometimes called a futures) is a standardized legal contract to buy or sell something at a predetermined price for delivery at a specified time in the future, between parties not yet known to each other. The asset transacted is usually a commodity or financial instrument. The predetermined price of the contract is known as the ''forward price''. The specified time in the future when delivery and payment occur is known as the ''delivery date''. Because it derives its value from the value of the underlying asset, a futures contract is a derivative. Contracts are traded at futures exchanges, which act as a marketplace between buyers and sellers. The buyer of a contract is said to be the long position holder and the selling party is said to be the short position holder. As both parties risk their counter-party reneging if the price goes against them, the contract may involve both parties lodging as security a margin of the value of the contract with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Texas At Austin
The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas. It was founded in 1883 and is the oldest institution in the University of Texas System. With 40,916 undergraduate students, 11,075 graduate students and 3,133 teaching faculty as of Fall 2021, it is also the largest institution in the system. It is ranked among the top universities in the world by major college and university rankings, and admission to its programs is considered highly selective. UT Austin is considered one of the United States's Public Ivies. The university is a major center for academic research, with research expenditures totaling $679.8 million for fiscal year 2018. It joined the Association of American Universities in 1929. The university houses seven museums and seventeen libraries, including the LBJ Presidential Library and the Blanton Museum of Art, and operates various auxiliary research facilities, such as the J. J. Pickle Researc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Swaption
A swaption is an option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term "swaption" typically refers to options on interest rate swaps. Types of swaptions There are two types of swaption contracts (analogous to put and call options): *A payer swaption gives the owner of the swaption the right to enter into a swap where they pay the fixed leg and receive the floating leg. *A receiver swaption gives the owner of the swaption the right to enter into a swap in which they will receive the fixed leg, and pay the floating leg. In addition, a "straddle" refers to a combination of a receiver and a payer option on the same underlying swap. The buyer and seller of the swaption agree on: *The premium (price) of the swaption *Length of the option period (which usually ends two business days prior to the start date of the underlying swap), *The terms of the underlying swap, including: **Notiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interest Rate Cap And Floor
An interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for each month the LIBOR rate exceeds 2.5%. Similarly an interest rate floor is a derivative contract in which the buyer receives payments at the end of each period in which the interest rate is below the agreed strike price. Caps and floors can be used to hedge against interest rate fluctuations. For example, a borrower who is paying the LIBOR rate on a loan can protect himself against a rise in rates by buying a cap at 2.5%. If the interest rate exceeds 2.5% in a given period the payment received from the derivative can be used to help make the interest payment for that period, thus the interest payments are effectively "capped" at 2.5% from the borrowers' point of view. Interest rate cap An interest rate cap is a derivative in which the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Futures Contract
In finance, a futures contract (sometimes called a futures) is a standardized legal contract to buy or sell something at a predetermined price for delivery at a specified time in the future, between parties not yet known to each other. The asset transacted is usually a commodity or financial instrument. The predetermined price of the contract is known as the ''forward price''. The specified time in the future when delivery and payment occur is known as the ''delivery date''. Because it derives its value from the value of the underlying asset, a futures contract is a derivative. Contracts are traded at futures exchanges, which act as a marketplace between buyers and sellers. The buyer of a contract is said to be the long position holder and the selling party is said to be the short position holder. As both parties risk their counter-party reneging if the price goes against them, the contract may involve both parties lodging as security a margin of the value of the contract with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bond Option
In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC. *A European bond option is an option to buy or sell a bond at a certain date in future for a predetermined price. *An American bond option is an option to buy or sell a bond ''on or before'' a certain date in future for a predetermined price. Generally, one buys a call option on the bond if one believes that interest rates will fall, causing an increase in bond prices. Likewise, one buys the put option if one believes that interest rates will rise. One result of trading in a bond option, is that the price of the underlying bond is "locked in" for the term of the contract, thereby reducing the credit risk associated with fluctuations in the bond price. Valuation Bonds, the underlyers in this case, exhibit what is known as pull-to-par: as the bond reaches its maturity date, all of the prices involved with the bond b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios. French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Margrabe's Formula
In mathematical finance, Margrabe's formula is an option pricing formula applicable to an option to exchange one risky asset for another risky asset at maturity. It was derived by William Margrabe (PhD Chicago) in 1978. Margrabe's paper has been cited by over 2000 subsequent articles. Google Scholar'"cites" page for this article/ref> Formula Suppose ''S1(t)'' and ''S2(t)'' are the prices of two risky assets at time ''t'', and that each has a constant continuous dividend yield ''qi''. The option, ''C'', that we wish to price gives the buyer the right, but not the obligation, to exchange the second asset for the first at the time of maturity ''T''. In other words, its payoff, ''C(T)'', is max(0, ''S1(T) - S2(T))''. If the volatilities of ''Si'' 's are ''σi'', then \textstyle\sigma = \sqrt, where ''ρ'' is the Pearson's correlation coefficient of the Brownian motions of the ''Si'' 's. Margrabe's formula states that the fair price for the option at time 0 is: :e^S_1(0) N( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Value Of Money
The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The time value of money is among the factors considered when weighing the opportunity costs of spending rather than saving or investing money. As such, it is among the reasons why interest is paid or earned: interest, whether it is on a bank deposit or debt, compensates the depositor or lender for the loss of their use of their money. Investors are willing to forgo spending their money now only if they expect a favorable net return on their investment in the future, such that the increased value to be available later is sufficiently high to offset both the preference to spending money now and inflation (if present); see required rate of return. History The Talmud (~500 CE) recognizes the time value of money. In Tractate Makkos page 3a the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forward Contract
In finance, a forward contract or simply a forward is a non-standardized contract between two parties to buy or sell an asset at a specified future time at a price agreed on at the time of conclusion of the contract, making it a type of derivative instrument.John C Hull'', Options, Futures and Other Derivatives (6th edition)'', Prentice Hall: New Jersey, USA, 2006, 3 The party agreeing to buy the underlying asset in the future assumes a long position, and the party agreeing to sell the asset in the future assumes a short position. The price agreed upon is called the ''delivery price'', which is equal to the forward price at the time the contract is entered into. The price of the underlying instrument, in whatever form, is paid before control of the instrument changes. This is one of the many forms of buy/sell orders where the time and date of trade is not the same as the value date where the securities themselves are exchanged. Forwards, like other derivative securities, can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
